Fourier Series
The (complex) Fourier series for f (x) dened for 0 x 2 is the periodic
function
an einx
fF (x) =
n=
where
1 2 inx
e
f (x) dx
2 0
using the orthogonality relation (m and n are integers)
an =
2
0
eimx einx dx = 2mn
Theorem. If f (x) is piecew
TRBDF2
R. E. Bank, W. M. Coughran, W. Fichtner, E. H. Grosse, D. J. Rose, and R. K. Smith, "Transient Simulation of Silicon Devices and Circuits," IEEE Transactions on Computer-Aided Design, CAD-4, 436451, 1985. M. J. Johnson and C. L. Gardner, "An Interf
MAT 421 Applied Computational Methods
Prof. Gardner ([email protected])
Reading: Section 1.7 (oating point) of Molers Numerical Computing with
MATLAB.
Homework 1
Due: Fri Jan 16
(1) Verify that the three-point central dierence formulas for f and f are
IEEE Floating Point (1985)
A real number r is represented on the computer by
r r = (1 + f ) 2n
where the 1 is a phantom. In 64-bit double precision oating point, there is
one sign bit s (s = 0 for + and s = 1 for ), the mantissa 0 f < 1 is
allotted 52 bit
Van der Pol Oscillator Equations
du
=v
dt
dv
= ( u2 )v u
dt
with initial conditions u(0) = 1, v(0) = 0. Take for example = 4 and
tf = 40.
Shaw Oscillator Equations
In the van der Pol equations, set u v and v u, and then add the
sinusoidal forcing term to
Equivalence Theorem (Lax-Richtmyer)
The Fundamental Theorem of Numerical Analysis. For consistent numerical
approximations, stability and convergence are equivalent.
A numerical method is consistent if the (global) error is proportional to tp ,
p > 0.
For
Polynomial Interpolation
Givenf0 , f1 , . . . , fN , nd a polynomial P (x) of degree N so that P (xi ) = fi .
This problem has a unique solution and
f (N +1) ()
x
f (x) P (x) =
(x x0 )(x x1 ) (x xN )
(N + 1)!
for some x0 x xN .
To show the polynomial is u
TRBDF2
R. E. Bank, W. M. Coughran, W. Fichtner, E. H. Grosse, D. J. Rose, and
R. K. Smith, Transient Simulation of Silicon Devices and Circuits, IEEE
Transactions on Computer-Aided Design, CAD-4, 436451, 1985.
M. J. Johnson and C. L. Gardner, An Interface
Numerical Methods for Initial Value Problems
Consider the IVP
du
= f (u), u(t = 0) = u0
dt
In one-step methods, we will approximate
()
du
un+1 un
f , un+1 un + t f
dt
t
where f is an approximation to the RHS of the IVP ().
Forward Euler
The forward Euler
Equivalence Theorem (Lax-Richtmyer)
The Fundamental Theorem of Numerical Analysis. For consistent numerical
approximations, stability and convergence are equivalent.
Lax proved for IVPs. The theorem applies as well to BVPs, approximations
to functions and
Numerical Methods for Boundary Value Problems
BVPs are usually formulated for y(x). Along the x axis, allocate gridpoints
xi , i = 0, . . . , N . BCs will be imposed at x0 and xN .
First and Second Derivative Matrices
First and second derivatives at the i
u(xi, tn) un, h x
i
The Heat/Diusion Equation
The diusion (or heat) equation is (D > 0)
2u
u
= D 2 , u(x, t = 0) = u0 (x)
t
x
The fundamental solution or kernel K(x, t) of the diusion equation is
K(x, t) =
1
x2
exp
4Dt
4Dt
which satises the initial val
Finding Roots
Strategy for nding f () = 0:
x
need a good rst guess (graph f (x)
bracket root if possible (guarantees convergence of bisection and false
position)
beware of vertical asymptotes!
tune method to problem: bisection (to get near root) + New
Derivative Approximations
Second-order accurate central dierence approx to rst derivative
df
dx
i
x2
fi+1 fi1
= fi +
f +
2x
6 i
First-order accurate backward dierence approx to rst derivative
df
dx
i
x
fi fi1
= fi
f +
x
2 i
First-order accurate forwa
Numerical Simulation of High Mach Number Astrophysical Jets
with Radiative Cooling
Carl Gardner
Arizona State University
Jeff Hester (ASU), Chi-Wang Shu (Brown), Youngsoo Ha (KAIST),
Steve Dwyer & Devon Powell (ASU), John Krist & Karl Staplefeldt
(JPL), K
Burgers Equation
Burgers equation
ut + uux = uxx
is the simplest PDE that models the more complicated Navier-Stokes equations (viscous uid dynamics, boundary layers, etc.). The inviscid Burgers
equation
1
ut + f (u)x = 0, f (u) = u2
2
is the simplest PDE
MAT 421 Applied Computational Methods
Prof. Gardner ([email protected])
Reading: Sections 7.7 (examples of IVPs) and 7.8 (Lorenz Equations) of
Molers Numerical Computing with MATLAB.
Homework 7
Due: Mon Mar 23
The parameters = 10, r = 28, and b = 8/3 a
MAT 421 Applied Computational Methods
Prof. Gardner ([email protected])
Reading: Sections 7.17.4 (numerical methods for IVPs) of Molers Numerical Computing with MATLAB.
Homework 6
Due: Wed Feb 25
(1) Prove that backward Euler is A-stable and L-stable.
MAT 421 Applied Computational Methods
Prof. Gardner ([email protected])
Reading: Sections 3.13.3 (interpolation) of Molers Numerical Computing
with MATLAB.
Homework 3
Due: Mon Feb 2
(1) Problem 3.3 in Moler. Use interpolate.m.
(2) Write down the lowest
MAT 421 Applied Computational Methods
Prof. Gardner ([email protected])
Reading: Sections 7.17.4 (numerical methods for IVPs) of Molers Numerical Computing with MATLAB.
Homework 5
Due: Mon Feb 16
(1) For du/dt = f (u), prove that the backward Euler met
MAT 421 Applied Computational Methods
Prof. Gardner ([email protected])
Reading: Sections 6.16.4 (numerical integration) of Molers Numerical
Computing with MATLAB.
Homework 4
Due: Mon Feb 9
(1) Derive Simpsons rule from the trapezoidal rule by applying